Evaluate the following limit: $$\lim_{\Theta \rightarrow 0}6\Theta \operatorname{cosec}(3\Theta) $$
My solution:
$$\eqalign{\lim_{\Theta \rightarrow 0}6\Theta \times \frac{1}{\sin(3\Theta) }&= \lim_{\Theta \rightarrow 0}6\Theta \times\frac{1}{\sin(3\Theta) }.\frac{3\Theta }{3\Theta }\\ &=\lim_{\theta\rightarrow 0}\frac{6\Theta}{3\Theta}\\ &=2}$$
Is this correct?
That's absolutely fine, but there's no need to multiply by $\frac{3 \Theta}{3\Theta}$.
Just skip that bit and note that, in the limit as $\Theta \to 0, \sin(3\Theta) \equiv3\Theta$, so $\lim\limits_{\Theta\to 0}\left[\frac{6\Theta}{\sin(3\Theta)}\right]= \require{cancel}\frac{6\cancel{\Theta}}{3\cancel{\Theta}} \equiv \boxed{2}$.